Approximation numbers of composition operators on the Hardy space of the ball and of the polydisk

We give general estimates for the approximation numbers of composition operators on the Hardy space on the ball $B\_d$ and the polydisk $D^d$

Approximation numbers of weighted composition operators

We study the approximation numbers of weighted composition operators $f\mapsto w\cdot(f\circ\varphi)$ on the Hardy space $H^2$ on the unit disc. For general classes of such operators, upper and lower bounds on their approximation numbers are derived. For the special class of weighted lens map composition operators with specific weights, we show how much the weight $w$ can improve the decay rate of the approximation numbers, and give sharp upper and lower bounds. These examples are motivated from applications to the analysis of relative commutants of special inclusions of von Neumann algebras appearing in quantum field theory (Borchers triples).Comment: 35 pages, no figures. Some typos removed, minor improvements in presentation, updated reference

Some new thin sets of integers in harmonic analysis

We randomly construct various subsets A of the integers which have both smallness and largeness properties. They are small since they are very close, in various senses, to Sidon sets: the continuous functions with spectrum in Λ have uniformly convergent series, and their Fourier coefficients are in ℓp for all p > 1; moreover, all the Lebesgue spaces LΛq are equal forq < +∞. On the other hand, they are large in the sense that they are dense in the Bohr group and that the space of the bounded functions with spectrum in Λ is nonseparable. So these sets are very different from the thin sets of integers previously known.On construit aléatoirement des ensembles Λ d'entiers positifs jouissant simultanément de propriétés qui les font apparaître à la fois comme petits et comme grands. Ils sont petits car très proches à plus d'un égard des ensembles de Sidon: les fontions continues à spectre dans Λ ont une série de Fourier uniformément convergente, et ont des coe fficients de Fourier dans ℓp pour tout p > 1; de plus, tous les espaces de Lebesgue LqΛ coïncident pour q < +∞. Mais ils sont par ail leurs grands au sens où ils sont denses dans le compactifi é de Bohr et où l'espace des fonctions bornées à spectre dans Λ n'est pas séparable. Ces ensembles sont donc très di fférents des ensembles minces d'entiers connus auparavant

Lp-valued measures without finite X-semivariation for 2 < p < ∞

We show that for 1 ≤ p < ∞, the property that every Lp-valued vector measure has finite X-semivariation in Lp(μ, X) is equivalent to the property that every continuous linear map from 1 to X is p-summing. For 2 < p < ∞, we explicitly construct an Lp([0, 1])-valued measure without finite Lp-semivariation.Generalitat ValencianaMinisterio de Educación y CienciaUniversidad Politécnica de ValenciaCentre for Mathematics and its Applications at the Australian National Universit

Function algebras with a strongly precompact unit ball

Let µ be a finite positive Borel measure with compact support K ⊆ C, and regard L∞(µ) as an algebra of multiplication operators on the Hilbert space L2(µ). Then consider the subalgebra A(K) of all continuous functions on K that are analytic on the interior of K, and the subalgebra R(K) defined as the uniform closure of the rational functions with poles outside K. Froelich and Marsalli showed that if the restriction of the measure µ to the boundary of K is discrete then the unit ball of A(K) is strongly precompact, and that if the unit ball of R(K) is strongly precompact then the restriction of the measure µ to the boundary of each component of C\K is discrete. The aim of this paper is to provide three examples that go to clarify the results of Froelich and Marsalli; in particular, it is shown that the converses to both statements are false.Ministerio de Educación, Cultura y DeporteJunta de Andalucí

Some revisited results about composition operators on Hardy spaces

We generalize, on one hand, some results known for composition operators on Hardy spaces to the case of Hardy-Orlicz spaces HΨ: construction of a “slow” Blaschke product giving a non-compact composition operator on HΨ; construction of a surjective symbol whose composition operator is compact on HΨ and, moreover, is in all the Schatten classes Sp(H2), p > 0. On the other hand, we revisit the classical case of composition operators on H2, giving first a new, and simplier, characterization of closed range composition operators, and then showing directly the equivalence of the two characterizations of membership in the Schatten classes of Luecking and Luecking and Zhu.Ministerio de Educación y Cienci

The canonical injection of the Hardy-Orlicz space HΨ into the Bergman–Orlicz space BΨ

We study the canonical injection from the Hardy-Orlicz space HΨ into the Bergman–Orlicz space BΨ..Ministerio de Ciencia e Innovació